Title:

Homomorphisms and derivations on Banach algebras

This thesis is concerned with some problems in three areas of Banach algebra theory. These are dealt with separately in Chapters 2, 3 and 4. Chapter 2 is concerned with certain automatic continuity problems for homomorphisms and derivations on Banach algebras. The main result is that if there exists a discontinuous homomorphism from a Banach algebra onto a semiprime Banach algebra, or a discontinuous derivation on a semiprime Banach algebra, then there exists a topologically simple radical Banach algebra. The main result of Chapter 3 is that there are no Jordan derivations which are not also associative derivations on any semiprime algebra over a field not of characteristic 2. It follows from this that every Jordan derivation on a semisimple Banach algebra is a derivation, and therefore continuous. The background to Chapter 4 is a theorem which states that if A is a C'algebra with identity, acted on by a group G of isometric automorphisms in such a way that A is Gabelian, then the mot of Ginvariant states of A is a simplex. This was proved by Lanford and Ruelle in connection with the C*algebra approach to statistical mechanics. Methods are developed to provide an alternative proof of this result and to investigate the possibility of similar results holding in special cases when A is not a C*algebra.
